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Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

Volume 189 / 2019

Kam Hung Yau Acta Arithmetica 189 (2019), 95-107 MSC: 11K60, 11L07, 11N36. DOI: 10.4064/aa180218-10-9 Published online: 23 April 2019

Abstract

For any $\varepsilon \gt 0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert \lt x^{-{1}/{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real numbers $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $\alpha$ is quadratic irrational then the asymptotic formulas hold for all sufficiently large $x$.

Our tools come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential sums.

Authors

  • Kam Hung YauDepartment of Pure Mathematics
    University of New South Wales
    Sydney, NSW 2052, Australia
    e-mail

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